Mathematics for Engineers B (Differential equations)

Course content

The students combine the learnt mathematical methods of multivariate calculus and differential equations in the description and investigation of applied problems in the engineering sciences. They use constructively the mathematical formalism of scalar and vector fields, of differential operators, of different integral concepts and of Fourier analysis to model and analyse mechanical applications. The students describe time-dependent processes by means of ordinary differential equations and explain the close relation to dynamics and to oscillations. They analyse the quantitative and qualitative behaviour of ordinary differential equations and explicate the basic existence and uniqueness theorems. The students model fundamental applications, derive the behaviour of the trajectories and calculate solutions of systems of differential equations manually as well as by use of modern computational tools. The students combine their competences in technical mechanics with those in mathematics and they transfer their detailed insight of the one-mass oscillator to more general oscillating systems and their motion. They identify the system response and transient parts of the oscillations, and they explain resonance phenomena.

Content

1. differential equations: conversion into systems of first order, slope field, modeling e.g. of an oscillator, solving ODEs with Mathematica and Matlab
2. simple solution procedures: separation of variables, ODEs in homogeneous variables, linear ODEs of first order, homogeneous and particular solution, variation transient and steady state, exact ODEs and integrating factor
3. existence and uniqueness: Peano existence theorem, Lipschitz continuity, Picard Lindelöf theorem
4. linear ODEs of n-th order: superposition principle, fundamental system, Wronski determinant and linear independence of solutions, variation of parameters
5. linear ODEs with constant coefficients: e-ansatz, harmonic oscillator, strongly and weakly damped oscillations, aperiodic limit case, system response to extrenal excitations including its derivation, resonance
6. systems of linear ODEs: e-ansatz, variation of constants, matrix notation
7. Laplace transform: properties of multiplication, derivative and damping, solving ODEs by Laplace transform discontinuous right-hand sides, Dirac’s delta-distribution and impact
8. boundary value problems: deformation of a string, Green's function
9. dynamical systems: Lotka-Volterra equations, phase plot, stationary, stable and asymptotically stable points